An asynchronous approach is proposed for replica control in distributed systems. This approach applies an extension of serializability called epsilon-serializability (ESR), a correctness criterion which allows temporary and bounded inconsistency in replicas to be seen by queries. Moreover, users can reduce the degree of inconsistency to the desired amount. In the limit, users see strict 1-copy serializability. Because the system maintains ESR correctness (1) replicas always converges to global serializability and (2) the system permits read access to object replicas before the system reaches a quiescent state. Various replica control methods that maintain ESR are described and analyzed. Because these methods do not require users to refer explicitly to ESR criteria, they can be easily encapsulated in high-level applications that use replicated data.